I want to solve this equation

$$
\partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0
$$

numerically.

I know that this equation can be transformed into the hypergeometric equation through the transformation
$$
\phi(\rho) = \rho^l (1+\rho^2)^{-\alpha} P(\rho)
$$
(in which $P$ is some function)
whose exact solution is the well known function see here
$$
_2 F_1(a,b;c;\rho) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{\rho^n}{n!}.
$$

The crucial characteristic of this function is that if $a$ or $b$ are negative integers, then the series is finite.

However, I'm interested in exploring a numerical solution for this equation and I would like to know how to obtain numerically the finite series solutions.

Any idea?

Thanks.

This post imported from StackExchange Mathematics at 2015-05-09 14:49 (UTC), posted by SE-user miguelFe